We prove the lemma through the induction on \(m\) . The case \(m=0\)  is given by (REF ). Now suppose the results hold up to \(k\le m-1\) .
In ([1]}), the time derivative of the Christoffel symbols \(\Gamma _{jk}^{i}\)  has the form \(A*\nabla A\) . Therefore we obtain,
\(\nabla _{\frac{\partial }{\partial t}}\nabla ^{m}A&=\nabla (\nabla _{\frac{\partial }{\partial t}}\nabla ^{m-1}A)+\nabla ^{m-1}A*A*\nabla A;\\&=\nabla \lbrace \nabla ^{m+1}A)+\sum \limits _{i+j+k=m-1}\nabla ^{i}A*\nabla ^{j}A*\nabla ^{k}A+\nabla ^{m-1}\tilde{R}(T,T)\rbrace \\&+\nabla ^{m-1}A*A*\nabla A;\\&=\nabla ^{m+2}A+\sum \limits _{i+j+k=m}\nabla ^{i}A*\nabla ^{j}A*\nabla ^{k}A+\nabla ^{m}\tilde{R}(T,T);\) 
